Time course of drug effect

Objectives

Learn how to perform a simultaneous fit of concentration
and effect data.

Use simulation to understand the properties of the turnover
model for delayed drug effect.

Introduction

The time course of drug effect can be described by linking
separate models for concentration and effect.

Immediate: Drug effects are determined by the concentration
in a compartment of the pharmacokinetic model.

Delayed:

Effect Compartment: Drug effects are determined by the
concentration in a hypothetical effect compartment whose input is from
a compartment of the pharmacokinetic model.

Physiological mediator: Drug effects are determined by
the concentration of a physiological mediator. The concentration of the
mediator is influenced by the drug concentration in one of 4 basic ways:

Decreased synthesis of mediator

Increased synthesis of mediator

Decreased elimination of mediator

Increased elimination of mediator

"It had long been believed that there is no relationship
between the drug concentration in plasma and time course of action for
many drugs..." (1)

We prescribe and administer drugs to produce effects.
Clinical preoccupation with merely what dose to give misses the point,
and assumes an easy equivalence between dose and effect. What effect
are we hoping to achieve, what concentration is this effect associated
with, and what dose will give this concentration? This kind of thinking
involves cognisance of the many variables known and unknown which can
influence each of these steps.

Dosing results in drug concentration. So achieving an
appropriate drug concentration is the first goal of drug
administration. However plasma concentration monitoring is only readily
available for a small number of drugs with low therapeutic index
including digoxin, theophylline, and a handful of antibiotics,
immunosuppressants, and anticonvulsants. In clinical practice adjusting
drug dosing to concentration can be fraught with difficulty due to
misconceptions about pharmacokinetics and lack of appreciation of
factors causing individual variation.

Drug doses are more commonly adjusted according to clinical
effect rather than concentration. This titration is more rapidly
accomplished in settings of more intensive clinical monitoring, such as
intensive care and anaesthesia. Titration to effect is most satisfying
to the clinician for drugs with faster apparent onset and offset of
effect, potentially resulting in instant gratification for the drug
administrator, and hopefully more effective targeting of drug response
for the patient.

Pharmacokinetics gives us drug concentration versus time,
while pharmacodynamics gives us drug effect versus concentration.
Concentration is the link between drug dosing and effect. Linking
pharmacokinetics and pharmacodynamics gives us drug effect versus time.
This is called the time course of effect.

Both the timing of onset and offset of drug effect may be
important. When will the effect start to be observed, when will the
effect peak, when will the effect be at steady state (where
applicable), and when will the effect decrease and then cease to be
observable? The Emax model is the most fundamental description of the
relationship between drug concentration and effect. This model is named
after the parameter Emax, which describes the maximum effect of a drug.
Since this implies effect at infinite drug concentration, Emax can
never be measured, but can only ever be estimated from the shape of the
response curve, approaching its asymptote.

Drugs work by having action on physiological systems. The
drug action produces a response in the physiological systems and
associated control mechanisms. These changes lead to an observed drug
effect. The unbound portion of the drug is responsible for its action,
however plasma concentration measures total concentration (both bound
and unbound).

Which effects are important? For ease of data collection
faster onset effects that are easy to measure are most often reported
scientifically and used for clinical titration of dose. However more
meaningful effects are often delayed and sometimes cumulative. For
example antihypertensives are used for cardiovascular disease risk
reduction where the goal effect is not just reduction in blood pressure
but reduction in myocardial ischaemia and stroke rates. The goal
effects of many drugs relate not to easily measurable short term
physiological change, but longterm reduction in morbidity and
mortality. There are many examples in intensive care medicine where a
focus on short term physiological change observed as drug effect does
not equate to long term beneficial outcome. For example early studies
on inotropic drugs in heart failure ( eg dobutamine) reported improved
haemodynamic parameters, which may be clinically insignificant, while
subsequent outcome studies demonstrated an increase or no effect on
mortality.

The timing of drug effects may be classified as immediate,
delayed, or cumulative. Very few drugs have immediate effects, heparin
being a rare example. Most drugs have a delayed effect. This delay may
be due to many different pharmacokinetic and pharmacodynamic factors eg
absorption after administration more peripherally; distribution and
transport to effect side eg (target organ, cell membrane, organelle),
receptor binding interactions, protein binding interactions, effects on
enzymes and other physiological mediators.

Three main causes of delay in time course of effect will be
explored: absorption, effect compartments, and indirect or
physiological substance mediated effects.

Absorption: Any drug that is not administered directly into a
central compartment usually has to be absorbed. Typically this is
described for the following routes of administration: oral, rectal,
subcutaneous, intramuscular, but may also include the systemic and
local effects of topical application such as transcutaneous,
transmucosal, conjunctival. Inhalational drugs are typically described
in terms of uptake rather than absorption, but the basic concept is
similar.

Absorption is complex process: it may involve diffusion down
concentration gradients, or osmotic gradient, or specific transport
factors. An absorption constant may be estimated to explain delays in
drug concentration and effect due to absorption.

Effect compartments: Theoretical effect compartments were
introduced to help explain the time delay between plasma concentration
and observed effect for some drugs. This delay may occur because the
effect site is not the central compartment, and hence time is required
for drug delivery to effect site, by perfusion, diffusion or transport.
At steady state plasma concentrations, then there will be a constant
rate of input into the effect compartment, so the time to steady state
effect site concentration will be determined by the rate of
equilibration half-life. Equilibration half-life is determined by
volume of distribution (organ size, tissue binding), and clearance
(blood flow, diffusion).

Physiological substance mediated effects: Drug effects may be
defined as immediate or delayed. In reality almost no drugs have a
truly immediate effect due to complex physiological control systems and
interactions that exist at baseline and after drug administration.
Drugs act typically at receptors, but the observed effect is only seen
later. Delayed effects may be due to drug effect on a physiological
mediator, often an enzyme system. A mediator is something which affects
a transition between one stage and another. It can be thought of as a
go between or intermediatory, occupying an intermediate position,
forming a connecting link between one thing and another.

Physiological mediators may be defined as physiological
substances that can be affected by a administered drug or physiological
process to produce an effect on another physiological process. A
mediator may be an enzyme, or other biologically active molecule, for
example nitric oxide.

Drugs that have an effect via a physiological mediator can be
modelled using delay due to the turnover of the mediator, hence these
models are known as turnover models. Time course of effect models link
pharmacokinetic and pharmacodynamic models to describe changes in
effect over time. Differential equations can be used to describe delays
in observed drug effect due to absorption, effect compartment
disposition and drug action via physiological mediators. Changes in
concentration over time are described by changes in rate in and changes
in rate out. In the case of drugs that work via physiological
mediators, the change in concentration of the mediator over time is
used as a monitor of drug effect.

Export the simulated ka1emaxc data from pkpd.xls to a
format that can be used by other programs

Create a new Excel workbook and add the following headings
in the top row. It is important to put the # before ID so that the same
file can be used for NONMEM.#ID TIME DV
DVID

Fill the ID column with the value 1 down to row 23.

Rows 2 to 12 will be populated with concentration observation
records. Copy
the
Time values from pkpd.xls into the TIME column and copy the Conc values
to the DV column in rows 2 to 12. Fill the DVID column with the value 1
for rows 2 to 11. The DVID value is used to identify the type of
observation (1=conc, 2=effect).

Rows 13 to 23 will be populated with effect observation
records. The ID remains as 1, because the data is from the same single
subject. Copy the Time values from pkpd.xls into the TIME column and
copy the Effect values to the DV column in rows 13 to 23. Fill the DVID column with the
value 2 for these rows.

Your worksheet should look similar to Table 1, but your DV values will
be slightly different.

A

B

C

D

1

#ID

TIME

DV

DVID

2

1

0

-0.1

1

3

1

0.25

1.7

1

4

1

0.5

2.5

1

5

1

0.75

3.5

1

6

1

1

4.1

1

7

1

1.5

5.1

1

8

1

2

5.4

1

9

1

3

5.1

1

10

1

4

4.4

1

11

1

6

2.6

1

12

1

8

1.5

1

13

1

0

-0.1

2

14

1

0.25

33.7

2

15

1

0.5

47.3

2

16

1

0.75

54.5

2

17

1

1

58.7

2

18

1

1.5

62.5

2

19

1

2

63.8

2

20

1

3

62.2

2

21

1

4

58.5

2

22

1

6

46.9

2

23

1

8

33.7

2

Table 1. Data file for ka1emaxc.

Now save the data file in "My Pharmacometrics Data\Time
Course of Effect" using 'Save As' and choose CSV (Comma delimited,
*.csv) format. Name the file ka1emaxc.csv.

The Phoenix program requires this kind of data in a less
flexible format. The concentration and effect observations need to be
in separate columns. You should move the effect observations and put
them in column D. Rename column C to 'CONC' and column D to 'EFFECT'.
Then delete rows 13 to 23. The worksheet should look like this:

A

B

C

D

1

#ID

TIME

CONC

EFFECT

2

1

0

-0.1

-0.1

3

1

0.25

1.7

33.7

4

1

0.5

2.5

47.3

5

1

0.75

3.5

54.5

6

1

1

4.1

58.7

7

1

1.5

5.1

62.5

8

1

2

5.4

63.8

9

1

3

5.1

62.2

10

1

4

4.4

58.5

11

1

6

2.6

46.9

12

1

8

1.5

33.7

Table 2. Phoenix data file for ka1emaxc.

Save the file as ka1emaxc_CE.csv in the "My Pharmacometrics
Data\Time Course of Effect" folder.

MONOLIX should identify that your input file has a header
row and your data should appear under the headings: ID TIME Y YTYPE. If
necessary, place a check in the 'Use header' box so that MONOLIX can
identify the data by your header row. Click accept.

Instead of using the model library, we will use MLXTRAN to
write out the ka1emaxc model. Start by opening a text editor (e.g.
Editplus) and enter the code shown in Figure 3. Note that the variable name "effect" cannot be used in MLXTRAN so use "eff".

;One compartment first order input and elimination, immediate Emax effect

IMPORTANT:
MLXTRAN is case sensitive. Take care to be consistent with upper and
lower case letters in names.

Create a "Time Course of Drug Effect\Monolix" folder and
save the model as ka1emaxc_mlxt.txt in the "Time Course of
Effect\Monolix" folder. You can create the Monolix folder using Windows
Explorer or from the Monolix Save project dialog box.

If
you get a compile error. Double check that your code is identical to
that shown in the Figure and be sure to press ENTER after the last line
of code - this is required to 'end' the last statement.

Change the initial parameter estimates under Fixed effects
to reasonable starting values based on the Excel parameters used for
simulation. Holding the mouse over each box will tell you, which
parameter the value is for.

Set the 'Stand. dev. of the random effects' to 0 for each
parameter by setting all the elements of 'The covariance model' to 0 by
clicking on any elements containing 1.

Because
these are data from a single individual, there is no between
subject variability (random effects). The SD of random effects is
therefore 0.

Set the residual error model to 'const' for both models.

Set the 'Residual error parameters' to 1 (SD of residual
error) for both types of observation (concentration, effect).

Click 'Check initial fixed effects'. A plot of predictions
based on the model and initial parameter estimates will display along
with the observed values. Close the 'Check initial fixed effects'
window.

When
you have more than one output you will need to select the 'y_1' or y_2'
output variable for each graph.

You
can visualise the effect of changing your parameter estimates by
adjusting the values in the bottom left of the window. When you have
chosen initial estimates that form a prediction that is similar to the
observations, click 'Set as initial values' to apply these values and
close the window.

IMPORTANT: Save the project
as ka1_emaxc_project.mat in your Time Course of Effect\Monolix folder.

Set the calculation options by ticking the 'Estimate the
population parameters', 'Estimate the Fisher Information Matrix' and
'Estimate the log-likelihood' boxes which are next to the 'Run' icon at
the top of the Monolix window). Make sure the other boxes are not checked especially 'Estimate the individual parameters'.

Estimate the parameters by clicking on Run. This will take
a while depending on the complexity of the model. During the estimation
process you can see how the parameter estimates are being changed and
settle down towards the final value.

When
the estimation finishes click on 'List' button below 'Graphics' at the
bottom of the Monolix window. Click to check both boxes (for Y1 conc
and Y2 effect outputs) for 'Individual Fits' and 'VPC' as Outputs.
Uncheck all the other types of plot.

Click on OK to close the 'Graphics' list window.

Then click on 'Display the Graphics' at the top of the Monolix window.

Look
at the Individual fits for Conc and Effect.You can select the graphic
plots by clicking on the tab at the bottom of the 'Figures' window.

Save a pdf copy of each plot in your Time Course of Drug Effect\Monolix project folder.

Look at the VPC plots for Conc and Effect.

Click
on Settings, use the default settings and add 'Individual Data'. Then
click on 'Bins and CI' at the bottom of the list of options. Click on
'Equal width' and then click on 'Display'. Save a pdf copy of each VPC
plot in your Time Course of Drug Effect\Monolix project folder.

View
the parameter estimates by clicking 'Last Results'. A text
file containing these results is saved in a 'pop_parameters.txt' in the
project folder.

Return to the Workflow ka1emaxc object and click on the 'Setup
Main (ka1emaxc)' icon.

If the source is not defined then drag the Data ka1emaxc object
to the Mappings window.

Click on the radio buttons to associate TIME with Time, CONC
with CObs, EFFECT with Eobs.

Click on the Initial Estimates tab and set appropriate
initial
parameter values. Change the plot variables to 'Conc v Time' (drop down
box under the parameter values) and 'Effect v Time' to see the observations and predictions. Confirm that the initial estimates
give a reasonable fit to the data.

Click on the 'Structure' tab and UN-check the 'Population?'
box (upper left) to perform an individual subject analysis..

Save the project with the name Time Course of Effect.phxproj
in a Time_Course_of_Effect\Phoenix folder.

Select the ka1emaxc icon in the Workflow (Object Browser
window). Click
on the Execute ka1emaxc icon at the top of the Phoenix window. This will
start the parameter estimation process.

Click on Output Data Theta icon to see the parameter
estimates.

Create a new Excel workbook in the Time Course of Effect\Phoenix folder called "Time Course of Effect
Results.xlsx".

Select
the Theta table values (including columns and rows) then copy and paste
the parameter estimates to
the Excel workbook.

Create a new Word document in the Time Course of Effect\Phoenix folder called "Time Course of Effect
Results.docx".

Copy the Excel parameter table and paste it into the Word
document.

Look at the Ind DV, IPRED vs IVAR plot then right click on
it
and copy a bitmap to clipboard. Paste the clipboard contents to the
Word
document.

Click on the ka1emaxc workflow object the press Ctrl-C to copy it.

Click on the Workflow object then press Ctrl-V to paste the ka1emaxc workflow object.

Rename the 'Copy of ka1emaxc' object to 'VPC ka1emaxc' and click on it to see its contents.

Click on the 'Structure' tab and check the 'Population?'
box (upper left) to peform a population analysis. This willl allow a VPC plot to be generated.

Set up a visual predictive check by clicking on the
Sim./Pred Check
button then click on Main and set # Replicates to 100, click on Binning
and check the None option, click on Quantiles and use the
values (5, 50, 95) for a 90% prediction interval, then click on
Quantile % and set the values to 2.5, 50 , 97.5 for a 95% confidence
interval

The
$OMEGA parameters are FIXed to 0. This is because there is only one
individual being modelled. These random effects parameters are used to
describe between subject variability when there is more than one
individual.

Save the file. Check using Windows Explorer that you can
find the file "ka1emaxc.ctl" in the User Defined Models\NONMEM folder.

The data file name must match in the $DATA record
of the ka1emaxc.ctl file.

In
order for NONMEM to find the data file the $DATA record has to include
a path relative to the folder where NONMEM is executed. This is why the
'..\..\' is put before the name of the data file which is located in
the Time Course of Effect folder.

Execute NONMEM with this command in the NONMEM window:

nmgo ka1emaxc

When
you get errors from NONMEM with the nmgo command then please read the
error message carefully and try to understand what it is telling you.
The usual errors that occur with these example problems will give you
some clues to what you might need to change in your ctl file.

Commands
can be recalled by using the Up arrow on your keyboard. You can easily
repeat the command without more typing or edit it to save the amount of
typing you do.

Use Excel to open the ka1emaxc.fit file in the ka1emaxc.reg results folder.

Select
column A then click on the "Data" menu item then click on "Text to
Columns". Click on "Finish". This will separate the values into
separate columns.

Select all cells in the worksheet (e.g. press
ctrl-A) then right click on a cell and click on "Format cells". Click
on the "General" option.. This will make the numeric values easier to
read.

Delete row 1 which contains the value "TABLE 1".

Click
on cell A1 then click on the "Data" menu item then click on "Data
Filter". This will let you choose rows with specific values of DVID by
clickiing on the arrow in the corner of the column with the heading
"DVID".

Use the Data Filter to select rows with DVID=1 (concentrations). Create a graph of TIME versus Y and DV.

Use the Data Filter to select rows with DVID=2 (effects). Create a graph of TIME versus Y and DV.

Save the Excel file with the graphs as ka1emaxc.xlsx in the Time Course of Effect\NONMEM folder.

Use EditPlus to look at the saved
results in the '.smr' file in the ka1emaxc.reg folder. The results are saved in a folder with the
same name as the ctl file but with the extension '.reg' e.g. use the
following command or open the file using the Windows Explorer.

Physiological mediator model

Turnover models are used to describe delayed responses due to
the turnover of a physiological mediator. The physiological mediator
(M) without drug is formed at a rate proportional to its initial
concentration (M0) and its elimination rate constant (kout). There are
4 basic ways that drugs can affect a turnover model. These give rise to
characteristic time courses of response.

Use Berkeley Madonna to open Pharmacometrics Data\Time
Course of Effect\turnover.mmd

Look at the model equations

Run the model and make a graph of Time vs Conc (left axis)
and Time vs M1, M2, M3, M4 (right axis)

Use the Parameters Batch Runs option to vary Dose from 10
to 1000 in 5 steps

Explain the pattern of times of peak effect shown for each
of the response curves (M1, M2, M3, M4)

Parameter Estimation

Write up the results of parameter estimation using MONOLIX, Phoenix
and NONMEM.

Describe the differences between the parameters used for
simulation and the parameter estimates obtained from MONOLIX, Phoenix and NONMEM.

Discuss how you might change the experimental design in
order to get better agreement between the simulation parameters and the
parameter estimates.